Ornstein–Uhlenbeck semi-groups on stratified groups

Abstract

We consider, in the setting of stratified groups G, two analogues of the Ornstein–Uhlenbeck semi-group, namely Markovian diffusion semi-groups acting on L q ( p d γ ) , whose invariant density p is a heat kernel at time 1 on G. Both generators have the same “carré du champ”. The first semi-group is symmetric on L 2 ( p d γ ) , with generator ∑ i = 1 n X i * X i , where ( X i ) i = 1 n is a basis of the first layer of the Lie algebra of G. The second one is compact on L q ( p d γ ) , 1 < q < ∞ , non-symmetric on L 2 ( p d γ ) and the formal real part of its generator N is ∑ i = 1 n X i * X i . The spectrum of N is the set of non-negative integers if polynomials are dense in L 2 ( p d γ ) , i.e. if G has at most 4 layers; we determine in this case the eigenspaces. When G is step two, we give another description of these eigenspaces, similar to the classical definition of Hermite polynomials by their generating function.